Question

Solve for x and y if log_2(x+y)=log_3(3x+4y)=3

Answer 1

To solve this problem, we will use the properties of logarithms.

First, using the property log_b(x) = log_c(x) / log_c(b), we can rewrite the equation log_2(x+y) = 3 as (x+y) = 2^3 = 8.

Next, using the property log_b(x) = log_c(x) / log_c(b), we can rewrite the equation log_3(3x+4y)=3 as (3x+4y) = 3^3 = 27.

So, we have the following system of equations:

x + y = 8 ...(1)
3x + 4y = 27 ...(2)

To solve this system, we can use the method of substitution.

Simplifying equation (1) to solve for x, we have:

x = 8 - y ...(3)

Substituting equation (3) into equation (2), we have:

3(8 - y) + 4y = 27

24 - 3y + 4y = 27

-y = 27 - 24

-y = 3

y = -3

Now substituting the value of y into equation (1), we have:

x + (-3) = 8

x - 3 = 8

x = 8 + 3

x = 11

Therefore, the solution to the system of equations is x = 11 and y = -3.